Gravity Bhecho Amplitudes Echo Amplitude One

The primary echo

When a gravitational wave merger is followed by a series of faint echoes, each bounce is weaker than the last. The first of those echoes, the one that arrives immediately after the main signal, serves as the reference point. In the Recognition Science framework, that primary echo is assigned an amplitude of exactly 1. The declaration echoAmplitude_one states this normalization: the amplitude at reflection number zero equals one. It is a definitional choice, not a measurement or a physical prediction.

The framework models each successive reflection as attenuated by a fixed factor, the reciprocal of the golden ratio, approximately 0.618. So the first echo has amplitude 1, the second about 0.618, the third about 0.382, and so on. The normalization at zero is what makes these ratios meaningful: every later amplitude is expressed relative to the primary echo. Without this anchor, the sequence of ratios would still hold, but the absolute scale would be arbitrary.

In Recognition Science, this fits a broader pattern where the golden ratio appears throughout the framework's structure. The amplitude sequence is one instance of that pattern. The declaration itself, however, only establishes the base value. It does not assert that any real black hole has produced such an echo, nor that the predicted ratio has been observed. Those are separate claims, tied to future gravitational wave data.

The theorem is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions. It is a small piece of a larger structure, but it plays a specific role: it fixes the starting point for a sequence that the framework predicts will follow a golden-ratio decay. The practical consequence is that any test of the framework's echo predictions must compare observed amplitudes against this normalized sequence.

theorem echoAmplitude_one : echoAmplitude 0 = 1 := by
  simp [echoAmplitude]

/-- Each echo is attenuated by 1/φ relative to the previous. -/
theorem echoAmplitude_succ_ratio (n : ℕ) :
    echoAmplitude (n + 1) = echoAmplitude n * phi⁻¹ := by
  unfold echoAmplitude
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  have : phi ^ (-((n : ℤ) + 1)) = phi ^ (-(n : ℤ)) * phi⁻¹ := by
    rw [show (-((n : ℤ) + 1)) = -(n : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((n + 1 : ℕ) : ℤ) = (n : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]

theorem echoAmplitude_one : echoAmplitude 0 = 1 := by
  simp [echoAmplitude]

What this page does not claim

No observed black hole echo has been confirmed to follow this amplitude sequence. The normalization does not fix the absolute energy scale of the echoes. This declaration does not address the time delay between successive echoes.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/BHEchoAmplitudes.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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